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Weak dependence of mixed moving average fields and ap- plications Based on joint work with Imma Curato and Bennet Str oh | October 9, 2019 | Institute of Mathematical Finance Robert Stelzer Page 2 Weak dependence of mixed moving average
Bennet Str¨
Based on joint work with Imma Curato and Robert Stelzer
Page 2 Weak dependence of mixed moving average fields and applications | Bennet Str¨
October 9, 2019
evy basis, (At)t∈R the σ-algebra generated by the set of random variables {Λ(B), B ∈ B(S ×(−∞, t])}.
θ-weakly dependent.
limit theorems.
suitable definition of weak dependence. Derive distributional limit theorems for such random fields.
Page 3 Weak dependence of mixed moving average fields and applications | Bennet Str¨
October 9, 2019
◮ F∗
u is the class of bounded functions from (Rn)u to R.
◮ Fu is the class of bounded, Lipschitz functions from (Rn)u to R. ◮ F =
u∈N∗ Fu and F∗ = u∈N∗ F∗ u .
◮ Lip(G) = supx=y
|G(x)−G(y)| x1−y1+...+xn−yn, where G : Rn → R.
Page 3 Weak dependence of mixed moving average fields and applications | Bennet Str¨
October 9, 2019
◮ F∗
u is the class of bounded functions from (Rn)u to R.
◮ Fu is the class of bounded, Lipschitz functions from (Rn)u to R. ◮ F =
u∈N∗ Fu and F∗ = u∈N∗ F∗ u .
◮ Lip(G) = supx=y
|G(x)−G(y)| x1−y1+...+xn−yn, where G : Rn → R.
Page 4 Weak dependence of mixed moving average fields and applications | Bennet Str¨
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Let X = (Xt)t∈R be an Rn-valued stochastic process. Then, X is called θ-weakly dependent if the θ-coefficients θ(h) = sup
u,v∈N∗ θu,v(h) −
→
h→∞ 0,
where
θu,v(h) = sup |Cov(F(Xi1, . . . , Xiu), G(Xj1, . . . , Xjv ))| F∞Lip(G) , F ∈ F ∗
u , G ∈ Fu,
(i1, . . . , iu) ∈ Ru, (j1, . . . , jv) ∈ Rv, i1 ≤ . . . iu ≤ iu + h ≤ j1 ≤ . . . ≤ jv
Under θ-weak dependence central limit theorems can be proven under slower decay of the weak dependence coefficient compared to η-weak dependence.
Page 4 Weak dependence of mixed moving average fields and applications | Bennet Str¨
October 9, 2019
Let X = (Xt)t∈R be an Rn-valued stochastic process. Then, X is called θ-weakly dependent if the θ-coefficients θ(h) = sup
u,v∈N∗ θu,v(h) −
→
h→∞ 0,
where
θu,v(h) = sup |Cov(F(Xi1, . . . , Xiu), G(Xj1, . . . , Xjv ))| F∞Lip(G) , F ∈ F ∗
u , G ∈ Fu,
(i1, . . . , iu) ∈ Ru, (j1, . . . , jv) ∈ Rv, i1 ≤ . . . iu ≤ iu + h ≤ j1 ≤ . . . ≤ jv
Under θ-weak dependence central limit theorems can be proven under slower decay of the weak dependence coefficient compared to η-weak dependence.
Page 5 Weak dependence of mixed moving average fields and applications | Bennet Str¨
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Let X = (Xt)t∈Rm be an Rn-valued random field. Then, X is called θ-weakly dependent if θ(h) = sup
u,v∈N∗ θu,v(h) −
→
h→∞ 0,
where θu,v(h) = sup |Cov(F(XΓ), G(X˜
Γ))|
F∞Lip(G) , F ∈ F∗, G ∈ F, Γ, ˜ Γ ⊂ Rm, dist(Γ, ˜ Γ) ≥ h, |Γ| ≤ u, |˜ Γ| ≤ v
There are no central limit theorems available achieving the weaker decay demands on the weak dependence coefficient under θ-weak dependence known from the process case.
Page 5 Weak dependence of mixed moving average fields and applications | Bennet Str¨
October 9, 2019
Let X = (Xt)t∈Rm be an Rn-valued random field. Then, X is called θ-weakly dependent if θ(h) = sup
u,v∈N∗ θu,v(h) −
→
h→∞ 0,
where θu,v(h) = sup |Cov(F(XΓ), G(X˜
Γ))|
F∞Lip(G) , F ∈ F∗, G ∈ F, Γ, ˜ Γ ⊂ Rm, dist(Γ, ˜ Γ) ≥ h, |Γ| ≤ u, |˜ Γ| ≤ v
There are no central limit theorems available achieving the weaker decay demands on the weak dependence coefficient under θ-weak dependence known from the process case.
Page 6 Weak dependence of mixed moving average fields and applications | Bennet Str¨
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Consider y = (y1, . . . , ym) ∈ Rm and z = (z1, . . . , zm) ∈ Rm. We say y <lex z if and only if y1 < z1 or yp < zp and yq = zq for some p ∈ {2, . . . , m} and q = 1, . . . , p − 1. Define the sets Vt = {s ∈ Rm : s <lex t} ∪ {t} and V h
t = Vt ∩ {s ∈ Rm : t − s∞ ≥ h} for h > 0.
1
1 2 3 4 5 6 7
Figure: Vt and V h
t for m = 2, h = 2 and t = (−2, 4)
Page 7 Weak dependence of mixed moving average fields and applications | Bennet Str¨
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Let X = (Xt)t∈Rm be an Rn-valued random field. Then, X is called θ-lex-weakly dependent if θlex
X (h) = sup u∈N∗ θu(h) −
→
h→∞ 0,
where θu(h) = sup |Cov(F(XΓ), G(Xj))| F∞Lip(G) , F ∈ F∗, G ∈ F, j ∈ Rm, Γ ⊂ V h
j , |Γ| ≤ u
Page 8 Weak dependence of mixed moving average fields and applications | Bennet Str¨
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Let (Dn)n∈N be a sequence of finite subsets of Zm with lim
n→∞ |Dn| = ∞ and
lim
n→∞
|Dn| |∂Dn| = 0. Consider the random quantity 1 |Dn|
1 2
Xj. What can we say about its asymptotic distribution?
Page 9 Weak dependence of mixed moving average fields and applications | Bennet Str¨
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Let X = (Xt)t∈Zm be a stationary centered real-valued random field such that E[|Xt|2+δ] < ∞ for some δ > 0. Assume that θlex
X (h) ∈ O(h−α) with α > m(1 + 1 δ).
Let σ2 =
k∈Zm E[X0Xk|I], where I is the σ-algebra of shift
invariant sets. Then 1 |Γn|
1 2
Xj d − − − →
n→∞ εσ,
with ε standard Gaussian, independent of σ2.
Page 10 Weak dependence of mixed moving average fields and applications | Bennet Str¨
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Let X = (Xt)t∈Rm be a random field, A = (At)t∈Rm ⊂ Rm a family
measure. Assume Xt to be measurable w.r.t. σ(M(B), B ∈ Bb(S × At)). Then, A is the sphere of influence and X an (A, M)-influenced random field. If A is translation invariant (At = t + A0), the sphere of influence is described by the set A0. We call A0 the initial sphere of influence. For m = 1 and At = Vt the above definition equals the class of causal processes.
Page 10 Weak dependence of mixed moving average fields and applications | Bennet Str¨
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Let X = (Xt)t∈Rm be a random field, A = (At)t∈Rm ⊂ Rm a family
measure. Assume Xt to be measurable w.r.t. σ(M(B), B ∈ Bb(S × At)). Then, A is the sphere of influence and X an (A, M)-influenced random field. If A is translation invariant (At = t + A0), the sphere of influence is described by the set A0. We call A0 the initial sphere of influence. For m = 1 and At = Vt the above definition equals the class of causal processes.
Page 10 Weak dependence of mixed moving average fields and applications | Bennet Str¨
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Let X = (Xt)t∈Rm be a random field, A = (At)t∈Rm ⊂ Rm a family
measure. Assume Xt to be measurable w.r.t. σ(M(B), B ∈ Bb(S × At)). Then, A is the sphere of influence and X an (A, M)-influenced random field. If A is translation invariant (At = t + A0), the sphere of influence is described by the set A0. We call A0 the initial sphere of influence. For m = 1 and At = Vt the above definition equals the class of causal processes.
Page 10 Weak dependence of mixed moving average fields and applications | Bennet Str¨
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Let X = (Xt)t∈Rm be a random field, A = (At)t∈Rm ⊂ Rm a family
measure. Assume Xt to be measurable w.r.t. σ(M(B), B ∈ Bb(S × At)). Then, A is the sphere of influence and X an (A, M)-influenced random field. If A is translation invariant (At = t + A0), the sphere of influence is described by the set A0. We call A0 the initial sphere of influence. For m = 1 and At = Vt the above definition equals the class of causal processes.
Page 11 Weak dependence of mixed moving average fields and applications | Bennet Str¨
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Let Λ be an Rd-valued L´ evy basis with characteristic quadruplet (γ, Σ, ν, π) and f : S × Rm → Mn×d(R) be a B(S × Rm)-measurable Λ-integrable function. Then, Xt =
is called a mixed moving average (MMA) field with kernel f.
Page 12 Weak dependence of mixed moving average fields and applications | Bennet Str¨
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Let (At)t∈Rm be a full dimensional, translation invariant sphere of influence with initial sphere of influence A0 ⊂ V0. This also implies At ⊂ Vt. If X is adapted to the σ-algebra generated by {Λ(B), B ∈ B(S × At)} it can be written as Xt =
f(A, t − s)Λ(dA, ds) =
f(A, t − s)✶A0(s − t)Λ(dA, ds).
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Let X be an (A, Λ)-influenced MMA field. Assume that
proper cone.
1 2 3 4 5
Figure: Aj and K for m = 2 and j = (−2, 1)
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Then, X is θ-lex-weakly dependent with coefficients θlex
X (h) ≤ 2 S
tr(f(A, −s)ΣΛf(A, −s)′)dsπ(dA) +
f(A, −s)µΛdsπ(dA)
2
, for all h > 0 and ΣΛ = Σ +
constant cK > 0.
Page 15 Weak dependence of mixed moving average fields and applications | Bennet Str¨
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Let (Xu)u∈Zm be a zero mean (A, Λ)-influenced MMA field with f ∈ L2+δ ∩ L2 for δ > 0,
a closed proper cone K ⊂ V0. Additionally assume θlex
X (h) = O(h−α), α > m(1 + 1 δ).
Then Σ =
k∈Zm E[X0X ′ k] is finite and
1 |Dn|
m 2
Xj d − − − →
n→∞ N(0, Σ).
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For a zero mean random field define Yj,k = XjXj+k − E[X0Xk], j ∈ Zm, k ∈ Nm. Consider the random quantity 1 |Dn|
m 2
Yj,k. What can we say about its asymptotic distribution?
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◮ (Xt)t∈Rm stationary with X ∈ Lp for p > 1. ◮ h : Rn → Rk, s.t. h(0) = 0 and for 1 ≤ a < p h(x) − h(y) ≤ cx − y(1 + xa−1 + ya−1), for x, y ∈ Rn, c > 0. ◮ If X is θ-lex-weakly dependent, then Yt = h(Xt) is θ-lex-weakly dependent with θlex
Y (h) = Cθlex X (h)
p−a p−1 ,
for all r > 0 and a constant C. ◮ Extend asymptotic results to higher sample moments.
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◮ (Xt)t∈Rm stationary with X ∈ Lp for p > 1. ◮ h : Rn → Rk, s.t. h(0) = 0 and for 1 ≤ a < p h(x) − h(y) ≤ cx − y(1 + xa−1 + ya−1), for x, y ∈ Rn, c > 0. ◮ If X is θ-lex-weakly dependent, then Yt = h(Xt) is θ-lex-weakly dependent with θlex
Y (h) = Cθlex X (h)
p−a p−1 ,
for all r > 0 and a constant C. ◮ Extend asymptotic results to higher sample moments.
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Let (Xu)u∈Zm be a zero mean (A, Λ)-influenced MMA field with f ∈ L4+δ ∩ L2 for δ > 0,
a closed proper cone K ⊂ V0. Additionally assume θlex
X (h) = O(h−α), α > m
δ
2+δ).
Then, Σk =
j∈Zm Cov(Y0,k, Yj,k) is finite and
1 |Dn|
m 2
Yj,k d − − − →
N→∞ N (0, Σk) .
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◮ Introduced in [Nguyen and Veraart, 2018]. ◮ Let A = (At(x))(t,x)∈R×Rm be an ambit set, i.e. At(x) ⊂ R × Rm−1 At(x) = A0(0) + (t, x), (Translation invariant) As(x) ⊂ At(x), s < t At(x) ∩ ((t, ∞)) × Rm−1 = ∅. (Non-anticipative) ◮ The (A, Λ)-influenced MMA field Yt(x) = ∞
exp(−λ(t − s))Λ(dλ, ds, dξ) is called MSTOU process.
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◮ Introduced in [Nguyen and Veraart, 2018]. ◮ Let A = (At(x))(t,x)∈R×Rm be an ambit set, i.e. At(x) ⊂ R × Rm−1 At(x) = A0(0) + (t, x), (Translation invariant) As(x) ⊂ At(x), s < t At(x) ∩ ((t, ∞)) × Rm−1 = ∅. (Non-anticipative) ◮ The (A, Λ)-influenced MMA field Yt(x) = ∞
exp(−λ(t − s))Λ(dλ, ds, dξ) is called MSTOU process.
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◮ In the following we consider a zero mean c-class MSTOU process, i.e. At(x) = {(s, ξ) : s ≤ t, x − ξ ≤ c|t − s|}. ◮ Assume
π(dλ, ds, dξ) = ds dξ f(λ)dλ.
1 2 3 4 5
Figure: c-class MSTOU process for c = 1
2 and (t, x) = (−2, 1)
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◮ Then, Yt(x) is θ-lex-weakly dependent with coefficients θlex
Y (h)≤
∞ (m − 1)! m−1
k=0 1 k!(2 λψ(h) c
)k (2λ)m e−2 λψ(h)
c f(λ)dλ
−✶{c>1}(2ψ(h))m−1 ∞ e−2 λψ(h)
c
− e−2λψ(h) 2λ f(λ)dλ 1
2
, Vm−1 is the volume of the m − 1-dimensional ball with radius c. ◮ Consider a Gamma(α, β) distributed mean reversion parameter λ (α > m and β > 0 ensure existence). ◮ The sample mean of Yt(x) is asymptotically normal if α > m
δ
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◮ Then, Yt(x) is θ-lex-weakly dependent with coefficients θlex
Y (h)≤
∞ (m − 1)! m−1
k=0 1 k!(2 λψ(h) c
)k (2λ)m e−2 λψ(h)
c f(λ)dλ
−✶{c>1}(2ψ(h))m−1 ∞ e−2 λψ(h)
c
− e−2λψ(h) 2λ f(λ)dλ 1
2
, Vm−1 is the volume of the m − 1-dimensional ball with radius c. ◮ Consider a Gamma(α, β) distributed mean reversion parameter λ (α > m and β > 0 ensure existence). ◮ The sample mean of Yt(x) is asymptotically normal if α > m
δ
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